Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 3275750, 12 pages

http://dx.doi.org/10.1155/2016/3275750

## Global Dynamics of a Compressor Blade with Resonances

^{1}Department of Mechanics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China^{2}Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China^{3}School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224051, China

Received 24 March 2016; Revised 16 June 2016; Accepted 12 July 2016

Academic Editor: Jaromir Horacek

Copyright © 2016 Xiaoxia Bian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The global bifurcations and chaotic dynamics of a thin-walled compressor blade for the resonant case of 2 : 1 internal resonance and primary resonance are investigated. With the aid of the normal theory, the desired form associated with a double zero and a pair of pure imaginary eigenvalues for the global perturbation method is obtained. Based on the simpler form, the method developed by Kovacic and Wiggins is used to find the existence of a Shilnikov-type homoclinic orbit. The results obtained here indicate that the orbit homoclinic to certain invariant sets for the resonance case which may lead to chaos in the sense of Smale horseshoes for the system. The chaotic motions of the rotating compressor blade are also found by using numerical simulation.

#### 1. Introduction

Compressor blades are widely used in many fields of aerospace, aeronautic engineering, and mechanical industry due to their excellent mechanical properties. The problem of nonlinear dynamics of the rotating blades had attracted lots of research interest during the past decade. Various strategies and approaches have been proposed for nonlinear dynamics of rotating blades (see, e.g., [1–15]). However, theoretical analysis of global dynamics of the rotating blades has not been concerned in the current available literature. Several researchers have examined the global behaviors of plates, beams, and belt (see, e.g., [16–22]), but the results cannot be directly extended to the case of rotating blades.

Yang and Tsao investigated the vibration and stability of a pretwisted blade under nonconstant rotating speed in [1], and they also predicted the time-dependent rotating speed leads to a system with six parametric instability regions in primary and combination resonances. Surace et al. [2] dealt with the coupled bending-bending-torsion vibration of rotating pretwisted blades. Şakar and Sabuncu [3] presented the static stability and the dynamic stability of an aerofoil cross section rotating blade subjected to an axial periodic force and took into account the effects of coupling due to the center of flexure distance from the centroid, rotational speed, disk radius, and stagger angle. Al-Bedoor and Al-Qaisia [4] used a reduced-order nonlinear dynamic model to research the steady-state response of the rotating blade under the main shaft torsional vibration. Tang and Dowell [5] analyzed the nonlinear response of a nonrotating flexible rotor blade subjected to periodic gust excitations theoretically and experimentally. They reported that there exists a periodic or possibly chaotic behavior in the blade. Choi and Chou [6] studied the dynamic response of turbomachinery blades with general end restraints by applying the modified differential quadrature method. A Monte Carlo approach was employed to explore a supercritical Hopf bifurcation and random bifurcation of a two-dimensional nonlinear airfoil in turbulent flow by Poirel and Price [7]. Lacarbonara et al. [8, 9] established the governing equations of the blades under the centrifugal forces and discussed linear modal properties and the nonlinear modes of vibration away from internal resonances, respectively. Yao et al. [10] performed a nonlinear dynamic analysis of the rotating blade with varying rotating speed under high-temperature supersonic gas flow; furthermore, they [11] explored the contributions of nonlinearity, damping, and rotating speed to the steady-state nonlinear responses of the rotating blade, and they also investigated the effects of the rotating speed on nonlinear oscillations of the blade. Wang and Zhang discussed the stability of a spinning blade having periodically time varying coefficients for both linear model and geometric nonlinear model and obtained the stability boundary of linear model and stability of steady-state solutions of nonlinear model in [12].

In many cases, blades are usually modeled as a pretwisted, presetting, thin-walled rotating cantilever beam because the shape of the blade is very complex. Many researchers carried out studies on the dynamic behavior of the beam of this kind and obtained a lot of valuable results (see, e.g., [13–15]). Several methods have been developed to research the global bifurcation behaviors and chaotic dynamics in nonlinear systems that possess homoclinic or heteroclinic orbits. There are three methods: Melnikov method, global perturbation method, and energy-phase method. Melnikov gave the condition under which a homoclinic orbit in the unperturbation system would break under perturbation and at last lead to chaos in the system. Based on Melnikov method, Wiggins studied the global behaviors of the three basic systems [23]. Then, Kovacic and Wiggins [24] developed the global perturbation method to present Shilnikov-type homoclinic orbit for resonant system. The energy-phase method proposed by Haller and Wiggins [25, 26] detected the existence of single-pulse and multipulse homoclinic orbits in a class of near Hamilton systems. Applying the latter two methods, there were many applications to investigate the global behaviors (see, e.g., [16–22]).

In this paper, we obtain a sufficient condition for the existence of Shilnikov-type homoclinic orbit of a compressor blade with 2 : 1 internal resonance and primary resonance using normal form theory and global perturbation method. Firstly, the formulas of the simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues are derived by normal form theory in Section 2. Then, the dynamics of unperturbed system and perturbed system are analyzed in Sections 3 and 4 in detail, respectively. The analysis indicate that Shilnikov-type homoclinic orbit exists in these cases. Finally, numerical simulations are given to confirm the result in Section 5 and the work ends in Section 6 with a short summary.

#### 2. Formulation of the Problem

A thin-walled compressor blade of gas turbine engines with varying speed under high-temperature supersonic gas flow is considered in [11]. It is modeled as a pretwisted, presetting, thin-walled rotating cantilever beam, considering the geometric nonlinearity, centrifugal force, the aerodynamic load, and the perturbed angular speed.

The pretwisted flexible cantilever blade, with length mounted on a rigid hub with radius , is considered [11]. It rotates at a varying rotating speed around its polar axis where , where is the rotating speed at the steady-state and is a periodic perturbation. It is also allowed to vibrate flexurally in the plane making an angle , as shown in Figure 1(a). The rotating blade is treated as a pretwisted, presetting, thin-walled rotating cantilever beam. The length and width of the cross section of the beam in the and directions are and , respectively, and the thickness of the thin-walled beam is . For the purpose of describing the motion of the rotating blade, different coordinate systems are needed. The origin of the rotating coordinate system is located at the blade root, and are the principal axes of an arbitrary beam cross section in the local coordinates (Figure 1(b)), and the transformations between two coordinate systems are shown as and , . is denoted as the pretwist at the beam tip; then, is the pretwist angle of a current beam cross section. The local coordinate system is defined on the cross section of the beam to describe the geometric configuration and the cross section, where and are the circumferential and thickness coordinate variables in Figure 1(c); the notion represents the points off the middle surface; it is different from the notion ; the relationship is and . Assume that and represent the displacements of an arbitrary point and a point in the middle surface of the rotating blades on the , , and directions, respectively. and represent the rotations about the - and -axis, respectively.